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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 136367.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
136367.m1 | 136367m3 | \([1, -1, 0, -733343, 241846226]\) | \(209267191953/55223\) | \(11509709095654847\) | \([2]\) | \(1228800\) | \(2.0663\) | |
136367.m2 | 136367m2 | \([1, -1, 0, -51508, 2794875]\) | \(72511713/25921\) | \(5402516514286969\) | \([2, 2]\) | \(614400\) | \(1.7197\) | |
136367.m3 | 136367m1 | \([1, -1, 0, -21863, -1207200]\) | \(5545233/161\) | \(33556003194329\) | \([2]\) | \(307200\) | \(1.3731\) | \(\Gamma_0(N)\)-optimal |
136367.m4 | 136367m4 | \([1, -1, 0, 156007, 19520584]\) | \(2014698447/1958887\) | \(-408275890865400943\) | \([2]\) | \(1228800\) | \(2.0663\) |
Rank
sage: E.rank()
The elliptic curves in class 136367.m have rank \(1\).
Complex multiplication
The elliptic curves in class 136367.m do not have complex multiplication.Modular form 136367.2.a.m
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.